# Polar Form Calculator

Omni's polar form calculator can help whenever you want to **convert a complex number from rectangular to polar form**. Scroll down to find a short article explaining what the different forms of a complex number are all about and what the **polar form conversion formulas** look like. Believe it or not, converting to polar form can be fun!

## What are the forms of a complex number?

The two main forms of a complex number are **polar** and **rectangular**. Usually, you first encounter the rectangular form `z = a + bi`

, where `a`

and `b`

are the **real part** and **imaginary part** of `z`

, respectively. They are both real numbers!

The **polar form** describes `z`

by `r × exp(φi)`

, where:

- The
**magnitude**`r`

is the distance from the origin`(0,0)`

to`z`

; and - The
**argument**`φ`

is the angle between the real axis (x-axis) and the radius connecting the origin`(0,0)`

and`z`

.

Look at the picture to better understand what the polar form is:

## How do I convert from rectangular to polar form?

To convert a complex number from the rectangular `a + bi`

to polar form `z`

, we use the formulas:

`r = √(a² + b²)`

and

`φ = atan2(b, a)`

.

If `a > 0`

, then `atan2(b, a) = arctan(b / a)`

. However, if `a < 0`

, then `atan(b /a)`

does not give us the correct angle - it must be corrected by `±π`

to send us into the correct quadrant of the plane. Formally `atan2(b, a)`

is defined as:

`atan(b / a)`

if`a > 0`

;`atan(b / a) + π`

if`a < 0 ≤ b`

;`atan(b / a) - π`

if`a,b < 0`

;`π/2`

if`a = 0 < b`

;`-π/2`

if`b < 0 = a`

; and- remains undefined if
`x = y = 0`

.

These are exactly the formulas behind Omni's polar form converter.

## How to use this polar form calculator?

To use this polar form converter, take a complex number in the rectangular form `a + bi`

and **input a and b into the respective fields** of our tool. The two ingredients of the polar form will appear immediately in their fields: the magnitude

`r`

and phase (argument) `φ`

. Take them and write down the polar form `r × exp(iφ)`

of your number.This can't get any easier!

## Omni tools for complex numbers

Complex numbers can find you in sooo many areas of science! Omni is well aware of that - we have built a whole collection of tools addressing different problems related to complex numbers. Make sure to check out at least a few of the following tools:

- Complex number calculator;
- a+bi form calculator;
- Multiply complex numbers calculator;
- Divide complex numbers calculator;
- Imaginary number calculator;
- Complex number to polar form calculator;
- Complex number to trigonometric form calculator;
- Complex number to rectangular form calculator; and
- i calculator.

## FAQ

### How do I convert from trigonometric to polar form?

To convert a complex number from trigonometric to polar form, you need to:

- Make sure your number is written as
`z = r × cos(φ) + r × i × sin(φ)]`

. - Extract the magnitude
`r`

and the argument`φ`

. - Write down your number as
`r × exp(φi)`

. - If in doubt, use an online polar form calculator.

### What is the polar form of -1?

The answer is `exp(πi)`

. To derive this answer, observe that the modulus (magnitude) of `-1`

is equal to `1`

. Next, we must determine `φ`

such that `cos(φ) = -1`

and `sin(φ) = 0`

. We easily verify that `φ = π`

satisfies these conditions. Hence, the whole number reads `1 × exp(πi)`

, as claimed.